Double integral: Stokes' theorem

Question

Honestly, I have no idea what Stokes' theorem is. I only know that the circulation can be found by this theorem. Would anyone mind helping me?

Answer 1

Stokes' theorem relates the circulation along a closed curve with the integral of a quantity known as "curl" over some surface where the given curve is the boundary, at each point multiplied with the unit normal vector of the surface.

In your case, the curve can be seen as a triangle contained in a plane with normal vector $(1,1,1)$, so if you normalise that you get your $\hat n$. The quantity in brackets is the cross product of the "vector" $$ \nabla = \left(\frac{d}{dx},\frac{d}{dy},\frac{d}{dz}\right) $$ with the vector field $\vec F$.

Answer 2

Stokes' theorem relates a surface integral (of a curl) to a line integral over the boundary of the surface. This means you can simplify some of the integrals by translating surface integral to a line integral or vice-versa. In essence, it's just a higher-dimensional version of the fundamental theorem of calculus: an integral of derivative (now curl) over an interval (now a surface) equals the value of the function (now a vector field) at the boundaries (now a line, so we still need another integral). See also the Gauss' theorem which does this for a 3D volume.

Now, just calculate the curl: $$\vec{F}=(z^2,y^2,x)$$ $$\nabla\times \vec{F}=(0,2z-1,0)$$ An integral ofer a triangle should be quite simple: you just need to figure out a normal (hint: it's along (1,1,1) due to symmetry) and choose appropriate coordinates to evaluate the integral.